It is enough to establish the factorisation for \(\omega \in {S}({\mathbb {R}}^n, \Lambda {\mathbb {R}}^n)\) . Indeed, since the Schwartz space of multivector fields is dense in \(L^{p}({\mathbb {R}}^n,\Lambda {\mathbb {R}}^n)\)  for \(1<p<\infty \) , and \(\mathcal {S},\mathbf {R}\text{\tiny {$\triangle $}}, \text{\tiny {$\triangle $}}\mathbf {R}\)  and \(\mathcal {R}\text{\scriptsize {$\triangledown $}}\)  are bounded on \(L^{p}({\mathbb {R}}^n,\Lambda {\mathbb {R}}^n)\) , the result follows by continuity. The functions \(x_\nu /\vert x\vert ^{n+1}\)  define tempered homogeneous distributions for each \(\nu \) . Hence, by acting component wise \(\mathcal {R}_{\nu }\omega \in {S}({\mathbb {R}}^n, \Lambda {\mathbb {R}}^n)\)  for every \(\omega \in {S}({\mathbb {R}}^n, \Lambda {\mathbb {R}}^n)\)  and consequently also \(\mathbf {R}\text{\tiny {$\triangle $}}\omega , \omega \text{\tiny {$\triangle $}}\mathbf {R},\mathbf {R}\text{\scriptsize {$\triangledown $}}\omega \in {S}({\mathbb {R}}^n, \Lambda {\mathbb {R}}^n)\) . Setting \(\eta (\xi )=\widehat{\mathcal {F}(\omega )(\xi )}\text{\tiny {$\triangle $}}i\xi /\vert \xi \vert \) , we apply Theorem 5 in [1]}
first to \(-\frac{i\xi }{\vert \xi \vert }\text{\tiny {$\triangle $}}\eta \)  and then to \(\eta (\xi )\)  and the result follows. The other formula follows similarly.
